ivc2003 (ref. 5)
TRANSCRIPT
Image analysis by bidimensional empirical mode decomposition
J.C. Nunes*, Y. Bouaoune, E. Delechelle, O. Niang, Ph. Bunel
Laboratoire d’Etude et de Recherche en Instrumentation, Signaux et Systemes (LERISS), Universite Paris XII-Val de Marne, Bat P2-Piece 230 61 Avenue du
General de Gaulle, 94010 Creteil Cedex, France
Received 30 September 2002; received in revised form 18 April 2003; accepted 9 May 2003
Abstract
Recent developments in analysis methods on the non-linear and non-stationary data have received large attention by the image
analysts. In 1998, Huang introduced the empirical mode decomposition (EMD) in signal processing. The EMD approach, fully
unsupervised, proved reliable monodimensional (seismic and biomedical) signals. The main contribution of our approach is to apply the
EMD to texture extraction and image filtering, which are widely recognized as a difficult and challenging computer vision problem. We
developed an algorithm based on bidimensional empirical mode decomposition (BEMD) to extract features at multiple scales or spatial
frequencies. These features, called intrinsic mode functions, are extracted by a sifting process. The bidimensional sifting process is
realized using morphological operators to detect regional maxima and thanks to radial basis function for surface interpolation. The
performance of the texture extraction algorithms, using BEMD method, is demonstrated in the experiment with both synthetic and
natural images.
q 2003 Elsevier B.V. All rights reserved.
Keywords: Bidimensional empirical mode decomposition; Texture analysis; Unsupervised texture decomposition; Radial basis function; Surface interpolation
1. Introduction
The joint space-spatial frequency representations have
received special attention in the fields of image processing,
vision and pattern recognition. Huang [15] introduces a
multiresolution decomposition technique: the empirical
mode decomposition (EMD), which is adaptive and appears
to be suitable for non-linear, non-stationary data analysis.
We propose a new analysis method of texture images based
on bidimensional empirical mode decomposition (BEMD),
firstly presented in Ref. [24].
This paper is organized as follow. Section 2 presents
state of the art of texture analysis methods. Section 3
presents an introduction to the EMD and its extension on
bidimensional data. It describes too the implementation
details of sifting process, including the extrema detection by
morphological reconstruction and the radial basis function
(RBF) for surface interpolation. In Section 4, experimental
results on different texture images indicate the interest of
this new multiresolution decomposition with statistical
descriptors. Finally, a conclusion is presented in Section 5.
2. Texture analysis methods
Texture features are determined by the spatial relations
between neighbouring pixels. The difficulty of texture
analysis is demonstrated by the number of different texture
definitions attempted by vision researchers [12,17,28].
Approaches to texture feature extraction and recog-
nition span a wide range of methods. Several books and
articles give overviews of the available methodology [9,
33]. There are four major issues in texture analysis [22]:
feature extraction (local texture properties), texture
discrimination (image partition corresponding to different
textures), texture classification (classes definition in finite
number, normal and pathological) and shape from texture
(reconstruction of 3D surface geometry from texture
information).
Four major method categories may be identified [20]:
statistical with cooccurrence and autocorrelation features
[12], geometrical with Voronoi tessellation [33] and
structural features, model based with Markov random field
[5,11] and fractal parameters [6,26,32], and multiscale
features with AM– FM analysis [13], morphological
operators [27], Wigner-Ville Distributions [14,31], Gabor
filters [8,16], and wavelet transforms [21,23].
0262-8856/03/$ - see front matter q 2003 Elsevier B.V. All rights reserved.
doi:10.1016/S0262-8856(03)00094-5
Image and Vision Computing 21 (2003) 1019–1026
www.elsevier.com/locate/imavis
* Corresponding author. Tel.: þ33-1-45-171-492.
E-mail address: [email protected] (J.C. Nunes).
A few comparisons between texture feature extraction
schemes have been presented in Refs. [22,29,30]. Com-
parative studies showed different conclusions. Different
setups, test images, and filtering methods may be the
reasons for the contradicting results. No single approach did
perform best or very close to the best for all images. The
multiresolution techniques, which based on human visual
perception, intend to transform images into a representation
in which both spatial and frequency information are present.
The most commonly multiscale features used are Wigner
distributions [14,31], Gabor functions [2,7,16] and wavelets
transforms [18,19,21,34,36]. These multiscale features try
to characterize textures by filter responses directly.
However, one difficulty of the multiresolution analysis is
its non-adaptive nature since it uses filtering schemes.
We have applied the EMD technique to texture images
for two reasons. The first, the advantage of this technique,
the EMD is a fully data driven method [25], does not use any
pre-determined filter [8] or wavelet functions [19]. The
second, we can easily implement a bidimensional extension
of EMD. This paper, which is based on multiscale
decomposition, examines the issue of designing texture
decomposition by BEMD.
3. Texture analysis based on the empirical mode
decomposition
3.1. The 1D empirical mode decomposition
In this paper, we use the EMD, first introduced by Huang
et al. [15]. Different applications as medical and seismic
signals analysis have showed the effectiveness of this
method. This method permits to analyse non-linear and non-
stationary data. Its principle is to decompose adaptively a
given signal into frequency components, called intrinsic
mode functions (IMF). These components are obtained from
the signal by means of an algorithm called sifting process.
This algorithm extracts locally for each mode the highest
frequency oscillations out of original signal.
3.1.1. Sifting procedure
The sifting procedure decomposes a sampled signal sðkÞ
by means of the EMD. The sifting procedure is based on two
constraints:
† each IMF has the same number of zero crossings and
extrema;
† each IMF is symmetric with respect to the local mean.
Furthermore, it assumes that s has at least two
extrema.
The EMD represent adaptively non-stationary signals as
sums of zero-mean AM–FM components [10], i.e. an IMF
is an AM–FM component. AM–FM analysis [13] has
been used successfully in a variety of applications including
non-stationary analysis, edge detection, image enhance-
ment, recovery of 3D shapes from texture, computational
stereopsis, texture segmentation and classification.
The sifting algorithm for s [ l2ðZÞ reads as follows:
1. Initialise: r0 ¼ s (the residual) and j ¼ 1 (index number
of IMF),
2. Extract the jth IMF:
3. (a) Initialise h0 ¼ rj21; i ¼ 1;
(b) Extract local minima/maxima of hi21;
(c) Compute upper envelope and lower envelope
functions xi21 and yi21 by interpolating, respect-
ively, local minima and local maxima of hi21;
(d) Compute mi21 ¼ ðxi21 þ yi21Þ=2 (mean envel-
ope),
(e) Update hi :¼ hi21 2 mi21 and i :¼ i þ 1;
(f) Calculate stopping criterion (standard deviation
SDji)
(g) Repeat steps (b) to (f) until SDji # SDMAX and put
then sj ¼ hi (jth IMF)
4. Update residual rj ¼ rj21 2 sj;
5. Repeat steps 2–4 with j :¼ j þ 1 until the number of
extrema in rj is less than 2.
3.1.2. Finding all the IMFs
1. Once the first set of ‘siftings’ results in an IMF, define
c1 ¼ h1i: This first component contains the finest spatial
scale in the signal.
2. Generate the residue, r1; of the image by subtracting out
c1; r1 ¼ I 2 c1: The residue now contains information
about larger scales.
3. Resift to find additional components r2 ¼ r1 2
c2;…; rn ¼ rn21 2 cn
The superposition of all the IMF reconstructs the data:
I ¼Pn
j¼1 ðcjÞ þ rn:
We have to determine a criterion for the sifting process
to stop. This can be accomplished by limiting the size
of the SD, computed from the two consecutive sifting
results as:
SD2ji ¼
XKk¼1
lðhjði21ÞðkÞ2 hjiðkÞÞl2
h2jði21ÞðkÞ
" #
3.2. The bidimensional empirical mode decomposition
Texture analysis is considered as a challenging task. The
ability to effectively classify and segment images based on
textural features is of key importance in scene analysis,
medical image analysis, remote sensing and many other
application areas. Feature extraction is the first stage of
image texture analysis. To extract the 2D IMF during the
sifting process, we have used morphological reconstruction
J.C. Nunes et al. / Image and Vision Computing 21 (2003) 1019–10261020
to detect the image extrema and RBF to compute the surface
interpolation. A 2D IMF is a zero-mean 2D AM–FM
component. The image AM–FM decomposition [13] is
partially unsupervised feature based segmentation algor-
ithm, whereas the EMD is fully unsupervised. In the two
methods, the results of the decomposition are very efficient
and competitive with the best analysis results [13,25].
We define a bidimensional sifting process [24]:
† Identify the extrema (both maxima and minima) of the
image I by morphological reconstruction based on
geodesic operators;
† Generate the 2D ‘envelope’ by connecting maxima
points (respectively, minima points) with a RBF;
† Determine the local mean m1; by averaging the two
envelopes;
† Since IMF should have zero local mean, subtract out the
mean from the image: I 2 m1 ¼ h1;
† repeat as h1 is an IMF.
As described above, the process is indeed like sifting to
separate the finest local mode from the image first based
only on the characteristic multiscale. The sifting process,
however, has two effects to eliminate riding waves and to
smooth uneven amplitudes.
3.2.1. Extrema detection
Morphological reconstruction is a very useful operator
provided by mathematical morphology [1]. Its use in
hierarchical segmentation proves its efficiency in all the
steps of the process, from extrema detection to hierarchical
image construction. The image extrema have been detected
by using morphological reconstruction based on geodesic
operators. We define the geodesic reconstruction as follows.
The grayscale reconstruction IrecIðJÞ of I from J is
obtained by iterating grayscale geodesic dilations ›n1 of J
under I until a stability is reached, i.e. IrecI ¼W
n$1 ›n1ðJÞ:
A well-known use of the morphological reconstruction is
the extraction of the extrema (minima and maxima). If we
take J ¼ I 2 1 (subtract one gray level to every pixel of
original image) and if we perform the reconstruction Irec
(by geodesic dilation) of J by I; the difference I 2 Irec
corresponds to the indicator function of the maxima of I
(Fig. 1).
Conversely, the difference between Irecp (reconstruction
by geodesic erosion) and I (original image) produces
the indicator function of the minima of I: This extrema
detection method is described in Ref. [35].
3.2.2. Surface interpolation by radial basis function
In Ref. [15], Huang proposed to use cubic spline
interpolation on non-equidistant sampled data. We have
choice to use the RBF rather than the bicubic spline for
different reasons developed in Ref. [4]. One of the technical
problems is that the cubic spline fitting creates distortions
near the end points. RBFs are presented as a practical
solution to the problem of interpolating incomplete surfaces
derived from three-dimensional (3D) medical graphics. A
RBF is a function of the form:
sðxÞ ¼ pmðxÞ þXNi¼1
liFðkx 2 xikÞ; x [ Rd; li [ R;
where
† s is the radial basis function,
† pm low degree polynomial, typically linear or quadratic, a
member of mth degree polynomials in d variables,
† k·k denotes the Euclidian norm,
† the li’s are the RBF coefficients,
† -F is a real valued function called the basic function,
† the xi’s are the RBF centres.
The radial basis approximation method offers several
advantages over piecewise polynomial interpolants. The
geometry of the known points is not restricted to a regular
grid. Also, the resulting system of linear equations is
guaranteed to be invertible under very mild conditions.
Finally, polyharmonic RBFs have variational characteriz-
ations, which make them eminently suited to interpolation
of scattered data, even with large data-free regions. These
applications include geodesy, geophysics, signal proces-
sing, and hydrology. RBFs have also been successful
employed for medical imaging and morphing of surfaces
in three dimensions.
Experimental results demonstrate that high-fidelity
reconstruction is possible from a selected set of sparse and
irregular samples. RBF are introduced to compute a
continuous surface through a set of irregularly spaced
points (extrema) during the sifting process.
4. Results and discussion
Real-world texture frame, as well synthetic texture
frames, has been used to test and validate the proposed
approach. The decomposition approach is applied to both
synthetic and natural images, created by composing textures
selected from Brodatz [3]. A study is performed to show the
efficiency and performance of the texture extraction. Then, a
set of texture images is selected to illustrate the application
procedure of the EMD method.Fig. 1. Morphological reconstruction Irec (by geodesic dilation) of I by
I 2 1:
J.C. Nunes et al. / Image and Vision Computing 21 (2003) 1019–1026 1021
4.1. Texture extraction
The 2D decomposition by sifting process of an image
provides a representation that is easy to interpret. Every
mode (IMF) contains information of a specific scale, which
is conveniently separated. Spatial information is retained
within the mode. Our algorithm is able to decompose a
broad range of texture elements. Examples using images
(200 £ 200 in grey scale) from the Brodatz texture album, or
using synthetic and real images are shown in Figs. 3, 6–12.
To stop the sifting process, we used the standard deviation
(SD). We have used SDMAX between 0.05 and 0.75.
In a first step, we applied our algorithm on a synthetic
image (Fig. 2a). It is an image of sinusoidal components,
built with a sum of three horizontal (h1 ¼ 40; h2 ¼ 14;
h3 ¼ 0:2) and three vertical (v1 ¼ 60; v2 ¼ 17; v3 ¼ 0:3)
frequencies with respective amplitudes (ah1 ¼ 40; ah2 ¼ 30;
ah3 ¼ 190; av1 ¼ 40; av2 ¼ 50; av3 ¼ 190). All stages of
results are shown in Figs. 2 and 3.
We observe the images of the minima (Fig. 2b), the
maxima (Fig. 2c) detection, the mean envelope (Fig. 2d) and
the subtraction from mean envelope (Fig. 2e). Fig. 3 shows
the performed decomposition in two modes (Fig. 3b and c)
and the residue image (Fig. 3d). To prove the effectiveness
of our method, we compare the density profiles of the
original image and the various frequencies composing it. In
Fig. 4, we traced the profile of density of sinusoidal images
(corner in top on the left towards the corner in bottom in
right-hand side): the sum of the three frequencies (Fig. 4a),
the high frequency (Fig. 4b), the medium frequency (Fig. 4c)
and the low frequency (Fig. 4d).
In Fig. 5, we can observe the density profile of
decomposition, the sum of three frequencies (Fig. 5a), the
first mode (Fig. 5b), the second mode (Fig. 5c) and the
image residue (Fig. 5d). We can compare these different
profiles. We observe that the first mode corresponds to the
sinusoidal image with the highest spatial frequency, the
second to the medium and the residue to the lowest. These
profiles are very similar. However, we can watch in residue
image (Fig. 5d) an irregularity, which is due to interpolation
distortions near the end points. In Fig. 6, the decomposition
is performed in two modes. This fabric texture is quite
regular, mainly diagonal and horizontal structures, making
it well suited to the proposed approach.
Fig. 3. Synthetic texture.
Fig. 4. Synthetic texture (frequency components).
Fig. 2. Photographic texture fabric.
J.C. Nunes et al. / Image and Vision Computing 21 (2003) 1019–10261022
The first mode corresponds to the woven structure, the
second to the pattern and the residue to the horizontal stripes
(black and white). In Fig. 7, the decomposition is performed
in three modes. It is interesting to observe that the first mode
corresponds to the woven structure, the second to the finest
pattern, the third to the medium pattern and the residue to
largest pattern.
In Fig. 8, the decomposition is performed in three modes.
In Fig. 9, the decomposition of MRI is performed in three
modes. We observe the three modes and the residue, which
contain the pattern structures from finest to coarsest.
We can observe that the extraction quality of a mode
depend on the quality of the previous modes. Therefore, the
stop criteria (SD) of sifting process is important.
4.2. Image filtering
Since sifting process extracts firstly the highest fre-
quency, the first modes correspond generally to the noise.
The information of the noise is contained in the very first
modes and in the residual image. Conversely, the image
tendency is contained only in the residual image.
Fig. 5. Synthetic texture (BEMD).
Fig. 6. Photographic fabric texture.
Fig. 7. Photographic texture fabric.
Fig. 8. Photographic texture (D69 Brodatz).
J.C. Nunes et al. / Image and Vision Computing 21 (2003) 1019–1026 1023
The tendency can be represented by a polynom of order
relatively low ð0; 1; 2Þ: The results in Fig. 10 show the
possibility of low level processing (filtering or denoising)
with this technique. After having applied the decompo-
sition, this filtering would be easily carried out by the
subtraction with the original image of one or several modes.
The residue represents the filtered image.
4.3. Extraction of inhomogeneous illumination
Since sifting process extracts firstly the highest fre-
quency, the firsts modes correspond generally to the noise.
Conversely, the image tendency is contained in the latest
mode or more generally in residual image resulting to the
sifting process (Fig. 11).
After the decomposition, we subtract residue image from
original image (Fig. 12c) to perform inhomogeneous
illumination correction.
4.4. Perspectives
In the bidimensional case, the regional extrema are not
always well defined. The saddle points (or more generally,
the pattern of ridges and valleys) should be taken into
account. Thus, we propose the straightforward extension of
the 1D EMD to the 2D case by using morphological
operators. To detect lines peaks (respectively, the line
Fig. 9. Brain MRI.
Fig. 11. Synthetic texture.
Fig. 10. Retinal image.
J.C. Nunes et al. / Image and Vision Computing 21 (2003) 1019–10261024
connecting the lowest points) corresponding to the maxima
(respectively, minima), we use the watershed, a powerful
tool for image segmentation. More details of the BEMD
from these lines (the saddle points) will be presented in
future paper.
We can observe in Fig. 12 the BEMD from these lines
and can compare these results with BEMD from regional
extrema in Fig. 9. For the BEMD from the regional extrema
(Fig. 9), we used SDMAX ¼ 0:75: For the BEMD from the
saddle points (Fig. 12), we used SDMAX ¼ 0:47: By
comparing Figs. 9 and 12, we can observe that the details
corresponding to the modes are finer for the BEMD from the
saddle points. Consequently, BEMD based on saddle points
seem to produced more IMF than the preceding version.
5. Conclusion
In this paper, we present a new modulation domain
feature-based approach for discriminating textured images.
We have applied the EMD to texture image analysis. The
BEMD permits to extract spatial frequency components or
Fig. 12. Brain MRI.
J.C. Nunes et al. / Image and Vision Computing 21 (2003) 1019–1026 1025
different spatial scales, i.e. structures from finest to coarsest
scales. This method, derived from the image data and fully
unsupervised, permits to analyse non-linear and non-
stationary data as texture images. We have shown
experimental results for both natural and synthetic textures.
By having access to these representations of scenes or
objects, we can concentrate on only one or several modes
(one individual or several spatial frequency components)
rather than the image entirety. Clearly the 2D EMD offers a
new and promising way to decompose and extract texture
features without parameter.
References
[1] S. Beucher, Geodesic reconstruction, saddle zones and hierarchical
segmentation, Image Anal. Stereol. 20 (2001) 137–141.
[2] A.C. Bovik, M. Clark, W.S. Geisler, Multichannel texture analysis
using localized spatial filters, IEEE Trans. Pattern Anal. Mach. Intell.
12 (1) (1990) 55–73.
[3] P. Brodatz, Textures: A Photographic Album for Artists and
Designers, Dover publications, New York, 1966.
[4] J.C. Carr, W.R. Fright, R.K. Beatson, Surface interpolation with radial
basis functions for medical imaging, Comput. Graph. Proc., Annu.
Conf. Ser. (SIGGRAPH 2001) (2001) 67–76.
[5] E. Cesmeli, D.L. Wang, Texture segmentation using Gaussian
Markov random fields and LEGION, Proc. IEEE Int. Conf. Neural
Networks (ICNN’97) III (1997) 1529–1534.
[6] C.C. Chen, J.S. Daponte, M.D. Fox, Fractal feature analysis and
classification in medical imaging, IEEE Trans. Med. Imaging 8 (1989)
133–142.
[7] J. Daugman, Complete discrete 2D Gabor transform by neural
networks for image analysis and compression, IEEE Trans. Acoust.,
Speech, Signal Process. 36 (1988) 1169–1179.
[8] D. Dunn, W.E. Higgins, J. Wakeley, Texture segmentation using 2-D
Gabor elementary functions, IEEE Trans. Pattern Anal. Mach. Intell.
16 (2) (1994) 130–149.
[9] K.B. Eom, Segmentation of monochrome and color textures using
moving average modeling approach, Image Vision Comput. 17 (3)
(1999) 231–242.
[10] P. Flandrin, G. Rilling, P. Goncalves, Empirical mode decomposition
as a filter bank, IEEE Signal Processing Letters (2003), accepted for
publication.
[11] G. Gimel’farb, Image Textures and Gibbs Random Fields, Kluwer
Academic Publishers, Dordrecht, 1999, 250 p.
[12] R. Haralick, Statistical and structural approaches to texture, IEEE
Proc. 67 (5) (1979) 786–804.
[13] J.P. Havlicek, D.S. Harding, A.C. Bovik, Multidimensional quasi-
eigenfunction approximations and multicomponent AM–FM models,
IEEE Trans. Image Process. 9 (2) (2000) 227–242.
[14] J. Hormigo, G. Cristobal, High resolution spectral analysis of images
using the pseudo-Wigner distribution, IEEE Trans. Signal Process. 46
(6) (1998) 1757–1763.
[15] N. Huang, et al., The empirical mode decomposition and the Hilbert
spectrum for non-linear and non-stationary time series analysis, Proc.
R. Soc., Lond. A 454 (1998) 903–995.
[16] A.K. Jain, F. Farrokhnia, Unsupervised texture segmentation using
Gabor filters, Pattern Recognition 24 (12) (1991) 1167–1186.
[17] B. Julesz, J.R. Bergen, Textons, the fundamental elements in
preattentive vision and perception of textures, The Bell Syst.
Technical J. 62 (6) (1983) 1619–1645.
[18] A. Laine, J. Fan, An Adaptive Approach for Texture Segmentation by
Multi-channel Wavelet Frames, Technical Report TR-93-025, Center
for Computer Vision and Visualization, University of Florida,
Gainesville, FL, August 1993.
[19] S. Livens, G. Van de Wouwer, Wavelets for Texture Analysis: An
Overview, In Proceedings of the Sixth International Conference on
Image Processing and its Applications (IPA’97), Dublin, Ireland,
1997, pp. 581–585.
[20] X. Liu, D. Wang, Texture classification using spectral histograms,
Electronic report 25 (OSU-CISRC-7/2000-TR17), citeseer.nj.nec.
com/liu00texture.html
[21] S. Mallat, Wavelets for a vision, Proc. IEEE 84 (4) (1996) 604–614.
[22] A. Materka, M. Strzelecki, Texture Analysis Methods: A Review.
Technical University of Lodz (1998), COST B11 Report.
[23] Y. Meyer, Wavelets: Algorithms and Applications, SIAM, Philadel-
phia, 1993.
[24] J.C. Nunes, Y. Bouaoune, E. Delechelle, S. Guyot, Ph. Bunel, Texture
analysis based on the bidimensional empirical mode decomposition,
Mach. Vision Appl. (2003) in press.
[25] P.J. Oonincx, Empirical mode decomposition: a new tool for S-wave
detection, CWI Rep. Probab., Networks Algorithms (PNA) PNA-
R0203 (2002) ISSN 1386-3711.
[26] A.P. Pentland, Fractal-based description of natural scenes, IEEE
Trans. Pattern Anal. Mach. Intell. 6 (1984) 661–674.
[27] R.A. Peters II, Morphological pseudo bandpass image decompo-
sitions, J. Electron. Imaging 5 (2) (1996) 198–213.
[28] R.M. Pickett, Visual Analyses of Texture in the Detection and
Recognition of Objects, Picture Processing and Psyeho Pietories,
Academic Press, New York, 1970.
[29] O. Pichler, A. Teuner, B.J. Hosticka, A comparison of texture feature
extraction using adaptive Gabor filtering pyramidal and tree structured
wavelet transforms, Pattern Recognition (29) (1996) 733–742.
[30] T. Randen, J.H. Husoy, Filtering for texture classification: a
comparative study, IEEE Trans. Pattern Anal. Mach. Intell. 21
(1999) 291–310.
[31] T.R. Reed, H. Wechesler, Segmentation of textured images and
Gestalt organization using spatial/spatial-frequency representations,
IEEE Trans. Pattern Anal. Mach. Intell. 12 (1990) 1–12.
[32] N. Sarkar, B.B. Chaudhuri, An efficient approach to estimate fractal
dimension of textural images, Pattern Recognition 25 (9) (1992)
1035–1041.
[33] M. Tuceryan, A.K. Jain, Texture analysis, in: C.H. Chen, L.F. Pau,
P.S.P. Wang (Eds.), The Handbook of Pattern Recognition and
Computer Vision, second ed., World Scientific, Singapore, 1998, pp.
207–248.
[34] M. Unser, Texture classification and segmentation using wavelet
frames, IEEE Trans. Image Process. 4 (11) (1995) 1549–1560.
[35] L. Vincent, Morphological grayscale reconstruction in image
analysis: applications and efficient algorithms, technical report 91-
16, Harvard Robotics Laboratory, November 1991, IEEE Trans.
Image Process. 2 (2) (1993) 176–201.
[36] M.V. Wickerhauser, Adapted Wavelet Analysis from Theory to
Software, IEEE Press, New York, 1994.
J.C. Nunes et al. / Image and Vision Computing 21 (2003) 1019–10261026